Textbook Question
84. Arc length for polar curves: Prove that the length of the curve r = f(θ) for α ≤ θ ≤ β is
L = ∫(α to β) √(f(θ)² + f'(θ)²) dθ.
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16. Parametric Equations & Polar Coordinates
Calculus in Polar Coordinates
4:13 minutesProblem 12.R.40b
Textbook Question
40–41. {Use of Tech} Slopes of tangent lines
b. Find the slope of the lines tangent to the curve at the origin (when relevant).
r =3 − 6 cos θ
Verified step by step guidance1
Recognize that the curve is given in polar form: \(r = 3 - 6 \cos \theta\). To find the slope of the tangent line at the origin, we need to express the curve in Cartesian coordinates or use the formula for the slope of a tangent line in polar coordinates.
Recall the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). The slope of the tangent line \(\frac{dy}{dx}\) can be found using the chain rule as \(\frac{dy/d\theta}{dx/d\theta}\).
Compute \(\frac{dr}{d\theta}\) by differentiating \(r = 3 - 6 \cos \theta\) with respect to \(\theta\). This gives \(\frac{dr}{d\theta} = 6 \sin \theta\).
Use the formulas for derivatives of \(x\) and \(y\) with respect to \(\theta\):
\(\frac{dx}{d\theta} = \frac{dr}{d\theta} \cos \theta - r \sin \theta\)
\(\frac{dy}{d\theta} = \frac{dr}{d\theta} \sin \theta + r \cos \theta\)
Evaluate \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\) at the value(s) of \(\theta\) where the curve passes through the origin (i.e., where \(r=0\)). Then find the slope of the tangent line as \(\frac{dy/d\theta}{dx/d\theta}\) at those \(\theta\) values.
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Video duration:
4mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates and Curves
Polar coordinates represent points using a radius and angle (r, θ) instead of Cartesian (x, y). Understanding how curves are defined in polar form, like r = 3 - 6 cos θ, is essential for analyzing their properties and converting between coordinate systems when needed.
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Intro to Polar Coordinates
Slope of Tangent Lines in Polar Coordinates
The slope of a tangent line to a polar curve at a given point can be found using the formula dy/dx = (dr/dθ sin θ + r cos θ) / (dr/dθ cos θ - r sin θ). This relates the derivatives of r with respect to θ to the slope in Cartesian terms, enabling the determination of tangent slopes at specific angles.
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05:13Slopes of Tangent Lines
Implicit Differentiation and Derivatives
Implicit differentiation is used to find derivatives when variables are interdependent, such as r and θ in polar equations. Calculating dr/dθ and applying the chain rule allows us to find the rate of change needed to determine the slope of tangent lines at points like the origin.
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05:14Finding The Implicit Derivative
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